Further versions of the convolution equation

Abstract

Many authors seem to be unaware of the importance or the existence of the earlier papers of Choquet and Deny [4] and Deny [6] on its convolution equation, even when these have links with their own papers. Lau and Rao [9] and Davies and Shanbhag [5] among others have established variants or extended versions of the results of Choquet and Deny [4] and Deny [5], and have given various applications of these. The recent monograph of Rao and Shanbhag [15] should provide the reader with the relevant details of the literature in this connection. In the present paper, we make some further observations on the literature on integral equations, pointing out, in particular, that certain results of Laczkovich [8] and Baker [3] are essentially simple corollaries to, or variants of corollaries to, the general theorem of Deny [6] or the theorem of Choquet and Deny [4]. In the process of doing this, we are led to some new versions of the existing results on integral equations.